A323128 Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 4, 0, 1, 18, 30, 0, 1, 74, 360, 360, 0, 1, 310, 3450, 8880, 6240, 0, 1, 1382, 31770, 160080, 271800, 146160, 0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080, 0, 1, 32398, 2918244, 42687960, 214527600, 468669600, 460474560, 166924800

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

Example

T(3,2) = 18: 1a|2a3b, 1a|2b3a, 1b|2a3b, 1b|2b3a, 1a3b|2a, 1b3a|2a, 1a3b|2b, 1b3a|2b, 1a2b|3a, 1b2a|3a, 1a2b|3b, 1b2a|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    4;
  0, 1,   18,     30;
  0, 1,   74,    360,     360;
  0, 1,  310,   3450,    8880,    6240;
  0, 1, 1382,  31770,  160080,  271800,   146160;
  0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080;
  ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
      *binomial(n-1, j-1)*A(n-j, k), j=1..min(k, n)))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[k!/(k - j)! Binomial[n - 1, j - 1]* A[n - j, k], {j, Min[k, n]}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-1 give: A000007, A057427.

Row sums give A104600.

Main diagonal gives A137341.

T(2n,n) gives A324523.

Sequence in context: A221817 A054375 A195739 * A334703 A259938 A342202

Adjacent sequences: A323125 A323126 A323127 * A323129 A323130 A323131

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 30 2019

Status

approved